Convergence to zero in C^∞ on a cylindrical end passes to the derivative

Let $F$ be a real normed space and $f:\mathbb{R}\times\mathbb{R}\to F$. If $f(s,t)\to0$ in $C^\infty$ as $s\to+\infty$, uniformly in $t$, then the derivative $Df:\mathbb{R}^2\to L(\mathbb{R}^2,F)$ also tends to $0$ in $C^\infty$.

Together with the companion statement for pairs, this lets $C^\infty$ convergence on a cylindrical end be fed into smooth nonlinear expressions in a function and its first derivatives.

Formalization Note TendstoZeroCInfty f means: $f$ is $C^\infty$ on some half-plane $(s_0,\infty)\times\mathbb{R}$, and for every $m$ the $m$-th Fréchet derivative $D^mf(s,t)$ tends to $0$ uniformly in $t$ as $s\to+\infty$.